Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever $E$ has a rational 3-isogeny. We also prove the analogous result for the sextic twists of $j$-invariant 0 curves. For a more general elliptic curve $E$, we show that the number of quadratic twists of $E$ up to twisting discriminant $X$ of analytic rank 0 (respectively 1) is $\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between $p$-adic logarithms of Heegner points and apply it in the special cases $p=3$ and $p=2$ to construct the desired twists explicitly. As a by-product, we also prove the corresponding $p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_{n}$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_{n}$, due to Vershynin (2011), is $2^{-n^{c}}$, for some unspecified small constant $c>0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_{n}$ is at most $2^{-n^{1/4}\sqrt{\log n}/1000}$ for all sufficiently large $n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing $p$, if the central critical $L$-value is zero then the $p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

Let $p(k)$ denote the partition function of $k$. For each $k\geqslant 2$, we describe a list of $p(k)-1$ quasirandom properties that a $k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\unicode[STIX]{x1D716}$ using $O(\unicode[STIX]{x1D70F}(\log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})))$ queries and $O(\unicode[STIX]{x1D70F}(\log ^{2}(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716}))n)$ additional 2-qubit gates, where $\unicode[STIX]{x1D70F}=d^{2}\Vert H\Vert _{\max }t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification relies on a new form of ‘oblivious amplitude amplification’ that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$, in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$-fold ${{\mathbb {A}}}^3$. We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$, that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$. Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$, where F is an equivariant exceptional locally free sheaf on a projective toric $3$-fold X.

As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$ , then

$$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$

$c>0$

$1/7$

$1/2$

$A+A+A$

$A\subset \{1,\ldots ,N\}$

$A\subset \mathbb{F}_{q}^{n}$

$A$

Hassett and Tschinkel gave counterexamples to the integral Hodge conjecture among 3-folds over a number field. We work out their method in detail, showing that essentially all known counterexamples to the integral Hodge conjecture over the complex numbers can be made to work over a number field.

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519–528] in which they were used to answer questions regarding point configurations. In this work, we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly’s theorem, which is the complex analog of the Sylvester–Gallai theorem.

We prove existence of solutions of the vacuum Einstein equations with initial data induced by a smooth metric on a light-cone.

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over  $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations $\mathsf{E}$ and $\mathsf{F}$. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation

$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$

$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$

$(1+1)$

We show that Bogoliubov theory correctly predicts the low-energy spectral properties of Bose gases in the Gross-Pitaevskii regime. We recover recent results from [6, 7]. While our main strategy is similar to the one developed in [6, 7], we combine it with new ideas, taken in part from [15, 25]; this makes our proof substantially simpler and shorter. As an important step towards the proof of Bogoliubov theory, we show that low-energy states exhibit complete Bose-Einstein condensation with optimal control over the number of orthogonal excitations.

We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$.